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\title{About Double Enneper}
\author{H. Karcher}
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   The surfaces Wavy Enneper,  Catenoid Enneper,  Planar Enneperand Double Enneper are finite total curvature minimal immersionsof the once or twice punctured sphere---shown with standardpolar coordinates. These surfaces illustrate how the differenttypes of ends can be combined in a simple way.
   
   The pure Enneper surfaces 
   ($\mathrm{Gauss}(z)=z^k,\  dh = \mathrm{Gauss}(z)\ dz$) 
  and the Planar Enneper surfaces \hfill\break
  \phantom{.}\hfill  
  ($\mathrm{Gauss}(z)=z^k,\  dh = \mathrm{Gauss}(z)/z^2\ dz$)
 \hfill\break have been re-discovered many times, because
the members of the associate family are \emph{congruent} surfaces
(as can be seen in an associate family morphing) and the
Weierstrass integrals integrate to polynomial (respectively)
rational immersions.  Double Enneper was one of the early
examples in which I joined two classical surfaces by a handle.
The Weierstrass data are:

Gauss map : $Gauss(z) = z^{ee-1}(z^{ee} - A^{ee})/(A^{ee}z^{ee}-1)$   
\hfill\break
Differential:  
  $ dh = e^{i\varphi}(1 - (z^{ee}-z^{-ee})/(A^{ee}-A^{-ee}))\, dz/z$.
  \hfill\break
  with $A = \sqrt{aa}\cdot\exp(i\alpha), \alpha = \pi bb/ee$  \hfill\break
  and   $ (A^{ee}+A^{-ee})\tan\varphi = - (A^{ee}-A^{-ee})\tan 2\alpha $.
  The last equality is needed to avoid periods of the Weierstrass integral.

  In this example the parameter aa  controls the size
of the neck between the top and bottom Enneper ends
(it should be kept in the range $3 < aa < 7$). The parameter
 bb rotates the top and the bottom ends relative to each other,
 The integer parameter $ ee = 2,3$,... determines the windingnumber of the Enneper ends and also the rotational symmetryof the surface---the default is $ee = 2$.
And umin and umax control how far into the ends one computes.

  Try a {\it Cyclic Morph} with the default parameters (this rotates
the upper and lower ends in opposite directions).  We also
suggest morphing the size (aa) of the handles. If the ends intersect
too much (e.g. for too large $u$-range or too large $ee$) one has
to reduce the $u$-range. The Cyclic Morph is also interesting for
large $ee$, for example $ee =12, umin=-1.45, umax=1.5$ and
reduced scaling.

  Formulas are taken from:

    H. Karcher, Construction of minimal surfaces, in "Surveys in
     Geometry", Univ. of Tokyo, 1989, and Lecture Notes No. 12,
     SFB 256, Bonn, 1989, pp. 1--96.


  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish Press, 1993

[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991



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